Beam deflection stresses

Replies to This Discussion

Permalink Reply by Robert Vier on November 19, 2012 at 7:00am

if you pre-bend the beam a bit and you apply a normal force, you should see a bending moment also without the large deformations module and without kangaroo...

Permalink Reply by Rasmus Holst on November 19, 2012 at 8:02am

Okay.... so I can not use the translational preDisp in one direction? It has to be a normal force?

If add rotation of 1 degree at the one point and then a normal force at the same point, the normal force has no effect on the bending moment. This is only affected by the rotation? Do I have to do something to take the column effect of something?

Permalink Reply by Robert Vier on November 23, 2012 at 5:46am

for me, with prescribedDispl I get bending:

what you cannot do is look at the section forces after you applied the large deformation module for a large prescribed displacement (e.g. you build a wooden gridshell by support movements) because the largeDeform component deletes the section forces after every step. this is a feature we are planning to implement, but it requires a bit more work (theory of large displacements)

Permalink Reply by Robert Vier on November 23, 2012 at 5:53am

the only (very rough and not nearly maintaining the requirements for linear FE) way to get a first figure is to displace the support of a slightly pre-bent beam (maybe an eccentricity of 1/20 of its length) and displace its support along the beam axis.

but still the resulting bending moment is very much depending on the initial imperfection (1/20  or 1/5 th ...) you applied.

see the attached file.

Attachments:

• karamba_preScrDispl_Bending_Test.gh, 12 KB

Permalink Reply by Daniel Piker on November 23, 2012 at 6:11am

Hi Rasmus,

If it is just bending and axial stresses you want, then I think I can output this from Kangaroo quite simply

(as the whole approach is completely based on large displacements from the start)

I'll try and post an example this weekend.

Permalink Reply by Robert Vier on November 23, 2012 at 6:40am

hi daniel,

how would you manage to get physical figures for bending and stress out of kangaroo?

I thought kangaroo is based on artificial mass-spring models.. have you found a way to implement materials, cross-sections etc?

best

Permalink Reply by Daniel Piker on November 23, 2012 at 8:46am

Hi Robert, nice to hear from you,

The member stresses are already calculated inside Kangaroo in order to find the accelerations of the nodes, and it is not such a big leap to output them.

I think there is often a bit of a misconception about the differences between 'mass-spring' models and FEA. Although the method of solving is different, as I do not form a global stiffness matrix, the elements themselves and the calculation of stresses in them can be effectively the same, and based on standard real material properties and sections.

Using nodes with only 3 degrees of freedom as Kangaroo does currently, axial stresses can be calculated (a spring being a very simple finite element), and bending without torsion (following the approach described in this paper), accounting for Young's modulus and sectional area. I had been focused for a while on more geometrical optimization, but recently have been looking again at clarifying the real world units and numerical values used by Kangaroo for structural purposes.

Several other ways of modelling beam/plate/volume type elements using combinations of springs are commonly used in game/animation physics, and these can indeed be difficult to link to accurate quantitative behaviour, which has perhaps helped form the impression of mass-spring models as non accurate, but it need not be so.

The approach can also be extended to 6dof nodes, in which case it becomes possible to include torsion, anisotropic bending etc, and to base these on more standard engineering formulations for beams and other elements.

In fact I've recently worked on some software together with Gennaro Senatore and Charlie Banthorpe for Expedition Workshed that implements such 6dof elements together with large displacements, realtime interaction, and options to output bending moment/shear/torsion graphically. This is browser based (you can try it here), rather than Grasshopper but I'm currently working on bringing the same approach into Kangaroo.

Maintaining interactive speeds while avoiding numerical instabilities does pose its challenges with these methods, and for many conventional structures where the displacements are small and interaction is less important I think conventional FEA will continue to be more efficient for some time, but I do believe the approaches will eventually converge.

Thinking about it - although they are very useful techniques, continuum mechanics and infinitesimal displacements are both just useful abstractions, and no less 'artificial' than mass-spring models (and I think infinitesimal displacements are particularly counter-intuitive - real things have to move to generate stresses).

Anyway, I'm always very interested in exploring collaborations and sharing of ideas about these approaches, and would love to hear any more thoughts from the Karamba team about this...

best,

Daniel

Permalink Reply by Karamba3D on November 23, 2012 at 11:02am

Hi Daniel, hi Rasmus,

the large deformation calculation in Karamba is approximate and intended for form-finding: Loads get applied in small increments. Each of these load-steps results in displacement increments which are used to update the geometry that serves as input for the next increment, and so on. It is a purely incremental procedure. The final displacements can be determined with reasonable accuracy - although there is an unavoidable drift of the calculated solution from the real behavior. Bending moments depend on second derivatives of the deflection. This entails much less accuracy as compared to the displacements. Therefore I decided not to output them.

A more correct procedure would be an incremental iterative scheme. After each load increment an iteration is performed which aims at equilibrium between internal and external forces. This resembles the explicit approach as applied in Kangaroo. The difference is only the determination of a stiffness matrix which results in fast convergence.  

I agree with Daniel that implicit and explicit finite element procedures both lead to physically meaningful results (with units and all). Due to the fact that explicit procedures do not depend on the formulation of a stiffness matrix they are popular for describing highly non-linear physical phenomena. In cases where it is possible to formulate a stiffness matrix however, implicit procedures win because they are computationally more efficient. 

Regarding the above beam example: In case one wants to calculate the bending moments one is currently better served with Kangaroo than with Karamba. For the deflections you can use both.

Best,

Clemens

Permalink Reply by Rasmus Holst on November 25, 2012 at 3:47am

Hi Robert and Clemens.

Thank you for your answers. 

I think the potential of working more smoothly together with Kangaroo would be great. But for now, do you think it would give me the somewhat correct results to calculate the strain from the curvature radius at each element and add this as a pretension (mm/m) or could you think of a better way? I am just thinking about eps=z/R. z being the position in the cross section and R the radius of the curvature circle.

Thanks Rasmus

Permalink Reply by Karamba3D on November 26, 2012 at 12:40am

Hi Rasmus,

you can take the position of the nodes, use them to construct a spline and take the splines radius of curvature R to calculate the bending moment which is M = -EI/R. 

Best,

Clemens

Permalink Reply by Rasmus Holst on November 26, 2012 at 11:18am

Hi Clemens.

Yes I know this, but would my approach - with finding the strain instead of the moment - and adding this as a pretension load give good results you think?

Thanks

Rasmus

Permalink Reply by Karamba3D on November 27, 2012 at 12:52am

Hi Rasmus,

I think you want to get the resulting bending moments, i.e. those from large deformations plus those from loads acting on the resulting arch. The former result from M=-EI/R, the latter can be obtained from a small deflection analysis on the deformed geometry.

Why do you want to know the resulting bending moments? Normally it's not material strength but deflections under working load that determine the design of predeformed structures like your arch above.

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